Measuring K0SK± interactions using Pb–Pb collisions at √sNN=2.76 TeV

.ALICE Collaboration

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received22May2017

Receivedinrevisedform24August2017 Accepted4September2017

Availableonline8September2017 Editor:L.Rolandi

WepresentthefirstevermeasurementsoffemtoscopiccorrelationsbetweentheK⁰_SandK^±particles.The analysis wasperformedonthedatafromPb–Pbcollisionsat√_s

NN=².76 TeVmeasuredbytheALICE experiment.Theobservedfemtoscopiccorrelationsareconsistentwithfinal-stateinteractionsproceeding via the a0(980) resonance. The extracted kaon source radiusand correlation strengthparameters for K⁰_SK⁻ are foundto beequal withintheexperimental uncertaintiesto thosefor K⁰_SK⁺.Comparing the results ofthe present studywith those from published identical-kaonfemtoscopic studies by ALICE, mass and coupling parameters for thea0 resonance are tested. Ourresults are also compatiblewith theinterpretationofthea0havingatetraquarkstructureinsteadofthatofadiquark.

©^{2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Identicalboson femtoscopy, especiallyof identicalcharged pi- ons, has been used extensively over the years to study experi- mentallythespace–timegeometryofthecollisionregioninhigh- energy particle and heavy-ion collisions [1]. Identical-kaon fem- toscopy studies have also been carried out, recent examples of which are the ones with Au–Au collisions at √

s_NN=^{200 GeV} by theSTARCollaboration[2](K⁰_SK⁰_S)andwithpp at√

s=^{7 TeV} andPb–Pbcollisions at√

s_NN=².76 TeV by theALICECollabora- tion [3–5] (K⁰_SK⁰_S and K^±K^±). The pair-wise interactions between the identical kaons that form the basis for femtoscopy are for K^±K^± quantum statistics and the Coulomb interaction, and for K⁰_SK⁰_S quantumstatisticsandthefinal-stateinteractionthroughthe

f₀(980)/a₀(980)thresholdresonances.

One can also consider the case of non-identical kaon pairs, e.g.K⁰_SK^± pairs.Besides thenon-resonantchannelswhichmaybe present, e.g. non-resonant elastic scattering or free-streaming of thekaonsfromtheirfreeze-outpositionstothedetector,theother only pair-wise interaction allowed for a K⁰_SK^± pair atfreeze out fromthecollisionsystemisa final-stateinteraction(FSI)through the a0(980) resonance. The other pair-wise interactions present for identical-kaon pairs are not presentfor K⁰_SK^± pairs because:

a) thereisnoquantumstatisticsenhancementsincethekaonsare notidentical,b)thereisnoCoulombeffectsinceoneofthekaons is uncharged,and c) thereis no strong FSIthrough the f0 reso-

E-mailaddress:alice-publications@cern.ch.

nancesincethekaonpairisinan I=^{1 isospinstate,asisthea}0, whereasthe f₀isan I=^{0 state.}

Another featureof the K⁰_SK^± FSI through thea₀ resonanceis, duetothea0 havingstrangenessS=^{0 andtheK0}_S ^beingalinear combinationoftheK⁰ andK⁰,

^K0S

= √

¹ 2

^K0

+

^K0

,

(1)

onlytheK⁰K⁺pairfromK⁰_SK⁺andtheK⁰K⁻pairfromK⁰_SK⁻have S=^{0 and thus can formthe a}0 resonance.This allows the pos- sibility tostudy theK⁰ and K⁰ sources separately since they are individually selected by studying K⁰_SK⁻ and K⁰_SK⁺ pairs, respec- tively.An additionalconsequence ofthisfeature isthat only50%

of eitherthe K⁰_SK⁻ or K⁰_SK⁺ detectedpairs will passthrough the a0 resonance.Thisistakenintoaccount intheexpressionforthe modelusedtofitthecorrelationfunctions.

On the other hand, the natural requirement that the source sizes extracted from the K⁰_SK^± femtoscopy agree with those ob- tained for the K⁰_SK⁰_S and K^±K^± systems allows one to study the propertiesofthea₀ resonanceitself.Thisisinterestinginitsown rightsincemanystudies discussthepossibilitythat thea₀,listed by the Particle Data Group as a diquark light unflavored meson state [6],could be a four-quark state,i.e.a tetraquark, ora “K–K molecule”[7–12].Forexample,theproductioncrosssectionofthe a0resonanceinareactionchannelsuchasK⁰K⁻→^a−₀ ^shouldde- pend on whetherthe a⁻₀ is composed ofdu or dssu quarks, the formerrequiringtheannihilationofthess pairandthelatterbe- ing a direct transfer of the quarks in the kaons to the a⁻₀. The http://dx.doi.org/10.1016/j.physletb.2017.09.009

0370-2693/©^{2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

resultsfromK⁰_SK⁻femtoscopymightbesensitivetothesetwodif- ferentscenarios.

InthisLetter, resultsfromthefirst studyof K⁰_SK^± femtoscopy arepresented.ThishasbeendoneforPb–Pbcollisionsat√

sNN= 2.76 TeVmeasured bytheALICEexperimentattheLHC[13].The physics goals ofthe present K⁰_SK^± femtoscopy studyare the fol- lowing:1) showtowhatextent theFSIthroughthea0 resonance describesthecorrelationfunctions,2) studytheK⁰ andK⁰ sources toseeiftherearedifferencesinthesourceparameters,and3)test publisheda0 massandcouplingparameters bycomparisonswith publishedidenticalkaonresults[5].

2. Descriptionofexperimentanddataselection

The ALICEexperiment andits performance in the LHC Run1 (2009–2013) are described in Ref. [13] and Ref. [14,15], respec- tively.About22×^{106Pb–Pbcollisioneventswith}0–10% centrality class takenin 2011were used in thisanalysis (the average cen- tralityinthis rangeis 4.9%due toa slighttrigger inefficiency in the 8–10% range). Events were classified according to their cen- tralityusing themeasuredamplitudesin theV0detectors,which consist of two arrays of scintillators located along the beamline andcoveringthe fullazimuth[16].Charged particleswererecon- structed and identified with the central barrel detectors located within a solenoid magnet with a field strength of B=⁰.5 T.

ChargedparticletrackingwasperformedusingtheTimeProjection Chamber(TPC) [17]andtheInnerTrackingSystem(ITS) [13].The ITSallowedforhighspatialresolutionindeterminingtheprimary (collision) vertex. Tracks were reconstructed and their momenta wereobtainedwiththeTPC. Amomentumresolutionoflessthan 10 MeV/c was typically obtainedfor the chargedtracks ofinter- est in this analysis. The primary vertex was obtained from the ITS,thepositionoftheprimaryvertexbeingconstrainedalongthe beamdirection(the“z-position”)tobewithin±^{10 cmofthecen-} teroftheALICEdetector.Inadditiontothestandardtrackquality selections, the track selections based on the quality of track re- constructionfitandthenumberofdetectedtrackingpoints inthe TPCwereusedtoensurethatonlywell-reconstructedtrackswere takenintheanalysis[14,15].

Particle identification (PID) for reconstructed tracks was car- ried out using both the TPCand the Time-of-Flight(TOF) detec- tor in the pseudorapidity range |

η

|<0.8 [14,15]. For each PID method,a valuewas assignedtoeachtrackdenotingthenumber of standard deviations between the measured track information andcalculatedvalues(Nσ)[5,14,15].ForTPCPID,aparametrized Bethe–Blochformulawasusedtocalculatethespecificenergyloss ^dE/dx^{inthedetectorexpectedforaparticlewitha givenmass} andmomentum.ForPIDwithTOF,theparticlemasswas usedto calculatetheexpectedtime-of-flightasafunction oftracklength and momentum. This procedure was repeated for four “particle species hypotheses”—electron, pion, kaon and proton—, and, for eachhypothesis,adifferentNσ valuewasobtainedperdetector.

2.1.Kaonselection

Themethodsusedto selectandidentifyindividual K⁰_S andK^± particlesarethesameasthoseusedfortheALICEPb–PbK⁰_SK⁰_S and K^±K^±analyses[5].Thesearenowdescribedbelow.

2.1.1. K⁰_Sselection

The K⁰_S particles were reconstructed from the decay K⁰_S →

π

⁺

π

⁻, with the daughter

π

⁺ and

π

⁻ tracks detected in the TPCandTOFdetectors.Pionswith pT>0.15 GeV/cwereaccepted (sinceforlower pT trackfindingefficiencydropsrapidly)andthe distance of closest approach to the primary vertex(DCA) of the

reconstructed K⁰_S was required to be less than 0.3 cm in all di- rections. The required Nσ values for the pions were NσT P C <3 and NσT O F <3 for p>0.8 GeV/c. An invariant mass distribu- tion for the

π

⁺

π

⁻ pairs was produced andthe K⁰_S was defined tobe resultingfromapairthat fellintotheinvariant massrange 0.480<mπ⁺π⁻<0.515 GeV/c².

2.1.2. K^±selection

Charged kaon tracks were also detected using the TPC and TOF detectors, and were accepted if they were within the range 0.14<pT<1.5 GeV/c. In orderto reduce the number ofsecon- daries(forinstancethechargedparticlesproducedinthedetector material, particles from weak decays, etc.) the primary charged kaon tracks were selected based onthe DCA, such that the DCA transverse to the beam direction was less than 2.4 cm and the DCA along the beam direction was lessthan 3.2 cm. If the TOF signal werenot available,therequired Nσ valuesforthecharged kaonswere NσT P C<2 for pT<0.5 GeV/c,andthetrackwas re- jectedforpT>0.5 GeV/c.IftheTOFsignalwerealsoavailableand p_T>0.5 GeV/c:NσT P C<3 andNσT O F<2 (0.5<p_T<0.8 GeV/c), NσT O F <1.5 (0.8<pT<1.0 GeV/c), NσT O F <1 (1.0< pT<

1.5 GeV/c).

K⁰_SK^± experimental pair purity was estimated from a Monte Carlo(MC)studybasedonHIJING[18] simulationsusingGEANT3 [19] tomodelparticletransport throughthe ALICEdetectors.The puritywas determinedfromthefractionofthereconstructedMC simulated pairs that were identified as actual K⁰_SK^± pairs input fromHIJING.Thepairpuritywasestimatedtobe88%forallkine- maticregionsstudiedinthisanalysis.

3. Analysismethods

3.1. Experimentalcorrelationfunctions

ThisanalysisstudiesthemomentumcorrelationsofK⁰_SK^±pairs usingthetwo-particlecorrelationfunction,definedas

C

(

k^∗

) =

A

(

k^∗

)/

B

(

k^∗

)

(2)

where A(k^∗)isthemeasured distributionofpairs fromthe same event, B(k^∗) is the reference distribution of pairs from mixed events,andk^∗ isthemagnitudeofthemomentumofeach ofthe particlesinthepairrestframe(PRF),

k^∗

=

(

s

−

^m2_K0

−

^m2_K±

)

²

−

^4m2_K0m²_K±

4s (3)

where,

s

=

^m2_K0

+

^m2_K±

+

^2EK⁰E_K±

−

²

^pK⁰

·

^pK^± (4) andm_K0 (E_K0)andm_K± (E_K±)aretherestmasses(total energies) oftheK⁰_SandK^±,respectively.

Thedenominator B(k^∗)was formedbymixingK⁰_S andK^±par- ticles from each eventwith particles from tenother events.The vertexesofthemixedeventswereconstrainedtobe within2cm of each other in the z-direction. A centrality constraint on the mixedeventswas foundnot tobe necessaryforthenarrowcen- tralityrange,i.e.0–10%,usedinthisanalysis.Correlationfunctions wereobtainedseparatelyfortwodifferentmagneticfieldorienta- tionsintheexperimentandtheneitheraveragedorfitseparately, dependingonthefittingmethodused(seebelow).

Correlationfunctionsweremeasuredforthreeoverlapping/non- exclusive pair transverse momentum (kT= |^pT,1+^pT,2|/2) bins:

all kT, kT<0.675 andkT>0.675 GeV/c.The meankT valuesfor thesethree binswere 0.675,0.425 and0.970 GeV/c,respectively.

Fig. 1.Examples of raw K⁰_SK⁺correlation functions for the threekT bins with linear fits to the baseline at largek^∗. Statistical uncertainties are shown.

Fig. 1showssamplerawK⁰_SK⁺correlationfunctionsforthesethree bins for one of the magnetic field orientations. One can see the mainfeatureofthefemtoscopiccorrelationfunction: thesuppres- sionduetothe strongfinal-stateinteractions forsmallk^∗.Inthe higherk^∗region,theeffectsofthea0 appeartonotbepresentand thuscouldbeusedasareference,i.e.“baseline”, forthea0-based model fitted to C(k^∗) in order to extract the source parameters.

Alsoshowninthefigurearelinearfitstothebaselineforlargek^∗. The effectson C(k^∗)by the a0 resonanceare mostly seen inthe k^∗<0.2 GeV/c region, wherethe widthof thea0 region reflects thesizeofthekaonsource(seeequationsbelow).

Correlationfunctionswerecorrectedformomentumresolution effects using HIJING calculations. HIJING was used to create two correlation functions: one interms of thegenerator-level k^∗ and one in terms of the simulated detector-level k^∗. Because HIJING doesnot incorporate final-stateinteractions, weights were calcu- lated using a 9th-order polynomial fit in k^∗ to an experimental correlation function and were used when filling the same-event distributions. These weights were calculated using k^∗. Then, the ratioofthe“ideal”correlationfunctiontothe“measured”one(for eachk^∗ bin) was multipliedto thedatacorrelation functionsbe- forethe fit procedure.This correction mostly affected thelowest k^∗ bins,increasingtheextractedsourceparametersbyseveralper- cent.

3.2. Final-stateinteractionmodel

The K⁰_SK^± correlation functions were fit with functions that include a parameterization which incorporates strong FSI. It was assumed that the FSI arises in the K⁰_SK^± channels due to the near-thresholdresonance, a0(980). This parameterization was in- troducedbyR.LednickyandisbasedonthemodelbyR.Lednicky andV.L.Lyuboshitz [20,21](see alsoRef. [2]for moredetails on thisparameterization).

Using an equal emission time approximation in the PRF [20], the elastic K⁰_SK^± transition is written as a stationary solution _−k∗(r^∗)ofthescatteringprobleminthePRF.Thequantityr^∗rep- resentstheemissionseparationofthepairinthePRF,andthe−^k∗ subscriptreferstoareversaloftimefromtheemissionprocess.At largedistancesthishastheasymptoticformofa superpositionof aplanewaveandanoutgoingsphericalwave,

_−k∗

(

r^∗

) =

^{e−ik∗·r∗}

+

^f

(

k^∗

)

^e

ik^∗r^∗

r^∗

,

(5)

where f(k^∗) is the s-wave K⁰K⁻ or K⁰K⁺ scattering amplitude whose contribution is the s-wave isovector a0 resonance (see Eq. (11) inRef.[2]),

Table 1

Thea0massesandcouplingparameters,allinGeV(takenfromRef.[2]).

Reference ma0 γa₀KK¯ γa0π η

Martin[7] 0.974 0.333 0.222

Antonelli[8] 0.985 0.4038 0.3711

Achasov1[9] 0.992 0.5555 0.4401

Achasov2[9] 1.003 0.8365 0.4580

f

(

k^∗

) = γ

_a

0→^KK m²_a0

−

s

−

i

( γ

_a

0→^KKk^∗

+ γ

a0→π ηk_{π η}

) .

(6)

In Eq.(6),ma0 isthe massof thea0 resonance, and

γ

_a

0→KK and

γ

a₀→π η are the couplings of the a0 resonance to the K⁰K⁻ (or K⁰K⁺) and

π η

channels, respectively. Also, s=⁴(m²_K0+^k∗2) and kπ η denotes the momentum in the second decay channel (

π η

) (seeTable 1).

The correlation function due to the FSI is then calculated by integrating_−k∗(r^∗)intheKoonin–Prattequation[22,23]

C

(

_k

^∗

) =

d³

r^∗S

(

r^∗

)

_−k∗

(

r^∗

)

²

,

(7)

where S(^r∗)isaone-dimensionalGaussiansourcefunctionofthe PRFrelativedistance^r∗withaGaussianwidthR oftheform

S

(

r^∗

) ∼

^{e−r∗2/(4R2)}

.

(8) Equation (7) can be integrated analytically for K⁰_SK^± correlations withFSIfortheone-dimensionalcase,withtheresult

C

(

k^∗

) =

¹

+ λ α

1

2

^f

(

k^∗

)

R

²

+

^2Rf

(

k^∗

)

√ π

R F1

(

2k^∗R

)

−

^If

(

k^∗

)

R F2

(

2k^∗R

) ,

(9)

where F1

(

z

) ≡

√ π

e^−z2erfi

(

z

)

2z

;

^F2

(

z

) ≡

¹

−

^e−z2

z

.

(10)

Inthe aboveequations

α

isthefractionofK⁰_SK^± pairs thatcome from the K⁰K⁻ or K⁰K⁺ system, set to 0.5 assuming symmetry in K⁰ and K⁰ production[2], R isthe radiusparameter fromthe sphericalGaussiansourcedistributiongiveninEq.(8),andλisthe correlation strength.Thecorrelation strengthis unityintheideal caseofpurea0-resonantFSI,perfectPID,aperfectGaussiankaon sourceandtheabsenceoflong-livedresonanceswhichdecayinto kaons. Notethat the formofthe FSIterm inEq.(9) differsfrom

theformoftheFSItermforK⁰_SK⁰_S correlations(Eq. (9) ofRef.[2]) byafactorof1/2 duetothenon-identicalparticlesinK⁰_SK^± cor- relationsandthusthe absenceoftherequirementto symmetrize thewavefunctiongiveninEq.(5).

As seen in Eq. (6), the K⁰K⁻ or K⁰K⁺ s-wave scattering am- plitude depends on the a0 mass and decay couplings. In the present work, we have taken the values used in Ref. [2] which have been extracted from the analysis of the a0 →

π η

spectra of several experiments [7–10], shown in Table 1. The extracted a₀ massanddecay couplingshave arange ofvalues forthe var- ious references. Except for the Martin reference [7], which ex- tracts the a0 values from the reaction 4.2 GeV/c incident mo- mentum K⁻+^p→⁺(1385)

π

⁻

η

using a two-channel Breit–

Wigner formula, the other references extract the a0 values from the radiative φ-decay data, i.e. φ→

π

⁰

ηγ

, from the KLOE col- laboration [24]. These latter three referencesapply a model that assumes, aftertakingintoaccount theφ→

π

⁰

ρ

⁰_→

π

⁰

ηγ

back- groundprocess,thattheφdecaystothe

π

⁰

ηγ

finalstatethrough theintermediate processesφ→^K+K−

γ

_→a0

γ

orφ→^K+K−→ a₀

γ

,i.e.the “chargedkaon loop model”[9].The main difference between these analyses is that the Antonelli reference [8] as- sumesa fixed a0 massin the fitof thismodel tothe

π

⁰

η

data, whereas the Achasov1 and Achasov2 analyses [9] allow the a0 mass to be a free parameter in the two different fits made to the data. It is assumed in the present analysis that thesedecay couplingswillalsobe validforK⁰K⁻ andK⁰K⁺ scatteringdueto isospininvariance. Correlationfunctions were fittedwithall four ofthesecases to see theeffect on theextracted source parame- ters.

3.3.Fittingmethods

Inordertoestimatethesystematicerrorsinthefittingmethod used to extract R and λ using Eq. (9), two different methods, judgedtobeequallyvalid,havebeenusedtohandletheeffectsof thebaseline:1) aseparate linearfit tothe “baselineregion,” fol- lowedby fittingEq.(9)tothecorrelationfunction dividedbythe linearfittoextractthesourceparameters,and2) acombinedfitof Eq.(9)andaquadraticfunctiondescribingthebaselinewherethe source parameters andthe parameters of the quadratic function arefittedsimultaneously.Thesourceparameters areextractedfor eachcasefrombothmethodsandaveraged,thesymmetricsystem- aticerrorforeachcaseduetothefittingmethodbeingone-halfof thedifferencebetweenthetwomethods.Bothfittingmethodswill nowbedescribedinmoredetail.

3.3.1. Linearbaselinemethod

Inthe“linearbaseline method,”forthe allk_T,k_T<0.675 and k_T>0.675 GeV/c bins the a₀ regions were taken tobe k^∗<0.3, k^∗<0.2 andk^∗<0.4 GeV/c,respectively.Inthehigherk^∗ region itwas assumedthat effects ofthe a0 were not presentandthus canbeusedasareference, i.e.“baseline”, forthea0-basedmodel fittedto C(k^∗), whichwas averaged over the two magnetic field orientationsusedintheexperiment, toextractthesourceparam- eters.ForthethreekTbins,linearfitsweremadeinthek^∗ ranges 0.3–0.45, 0.2–0.45 and 0.4–0.6 GeV/c, respectively, and the cor- relation functionswere divided by these fits to remove baseline effectsextendingintothelow-k^∗ region.Theserangesweretaken to define the baselines since the measured correlation functions werefound tobe linearhere. Forlargervalues ofk^∗ thecorrela- tionfunctionsbecamenon-linear.Thebaseline was studiedusing HIJINGMCcalculationswhichtakeintoaccountthedetectorchar- acteristics as described earlier. The C(k^∗) distributions obtained fromHIJINGdonotshowsuppressionsatlowk^∗ asseen inFig. 1

but rathershow linear distributions over the entire ranges ink^∗ showninthefigure.HIJINGalsoshowsthebaselinebecomingnon- linear for larger values ofk^∗, asseen in the measurements. The MCgeneratorcodeAMPT[25]wasalsousedtostudythebaseline.

AMPT issimilar to HIJING butalso includes final-state rescatter- ing effects.AMPT calculationsalsoshowedlinearbaselinesinthe k^∗ ranges used in the present analysis, becoming non-linear for largerk^∗.Both HIJINGandAMPT qualitatively showthe samedi- rectionofchangesintheslopesofthebaselinevs.kTasseeninthe data,butAMPT moreaccurately describedtheslopevaluesthem- selves, suggestingthat final-staterescattering plays a role in the k_T dependenceofthebaselineslope.Thesystematicuncertainties ontheextractedsourceparametersduetotheassumptionoflin- earity inthesek^∗ regions were estimatedfromHIJING tobe less than 1%.

Fig. 2 shows examples of K⁰_SK⁺ and K⁰_SK⁻ correlation func- tions dividedby linearfits to thebaseline withEq.(9)using the Achasov2 parameters. One can seethe main feature ofthe fem- toscopic correlation function: the suppression due to the strong final-stateinteractionsforsmallk^∗.Asseen,thea0 FSIparameter- izationgives an excellent representationofthe “signalregion” of thedata,i.e.thesuppressionofthecorrelationfunctionsinthek^∗ range0toabout0.15GeV/c.

3.3.2. Quadraticbaselinemethod

Inthe “quadraticbaseline method,” R andλ areextractedas- suming a quadratic baseline function by fitting the product of a quadraticfunction andtheLednicky equation, Eq.(9),tothe raw correlation functions for each of the two magnetic field orienta- tionsusedintheexperiment,suchasshowninFig. 1,i.e.,

C_raw^{f it}

(

k^∗

) =

^a

(

1

−

^bk∗

+

^ck∗2

)

C

(

k^∗

)

(11) where C(k^∗) is givenby Eq.(9), anda, b and c are fit parame- ters.Eq.(11) isfit tothe samek^∗ ranges asshowninFig. 1,i.e.

0–0.45 GeV/cforallkTandkT<0.675 GeV/c,and0–0.6 GeV/cfor kT>0.675 GeV/c. The fits to the experimental correlation func- tionsarefoundtobeofsimilargoodqualityasseenforthelinear baselinemethodfitsshowninFig. 2.

3.4. Systematicuncertainties

Systematic uncertainties on the extracted source parameters were estimated by varying the ranges of kinematic and PID cut valuesonthedataby±^10%and±^{20%,aswell asfromMCsimu-} lations.Themainsystematicuncertainties ontheextractedvalues of R andλ duetovarioussources, notincludingthebaseline fit- tingmethod,are:a)k^∗ fittingrange:2%,b)single-particleandpair cuts (e.g. DCAcuts, PID cuts, pair separation cuts): 2%–4% for R and3%–8% for λ,andc)pairpurity:1%onλ.Combiningtheindi- vidualsystematicuncertaintiesinquadrature,thetotalsystematic uncertaintiesontheextractedsourceparameters,notincludingthe baseline fittingmethodcontribution,are inthe ranges3%–5% for Rand4%–8% forλ.

As mentioned earlier, for the two fitting methods, the source parametersareextractedforeachcasefrombothmethodsandav- eraged, the symmetric systematicerror foreach casedue to the fitting methodbeing one-half ofthe difference betweenthe two methods. The baseline fitting method systematic error thus ob- tained is added in quadrature with the systematic errors given above.Itisfoundthatthesizeofthebaselinefittingmethodsys- tematicerrorsareabout50%largerforRandofsimilarmagnitude forλasthosequotedaboveforthenon-fitting-methodsystematic errors.

Fig. 2.ExamplesofK⁰_SK⁺ and K⁰_SK⁻ correlationfunctionsdividedbylinearfitstothebaselinewiththeLednickyparameterizationusingthe Achasov2[9]parameters.

Statistical(lines)andthelinearsumofstatisticalandsystematicuncertainties(boxes)areshown.

4. Resultsanddiscussion

Fig. 3 shows sample results for the R and λ parameters ex- tracted in the present analysisfrom K⁰_SK^± femtoscopyusing the Achasov1parameters. The left columncompares K⁰_SK⁺ andK⁰_SK⁻ resultsfromthequadratic baselinefit method,andtherightcol- umn compares results averaged over K⁰_SK⁺ and K⁰_SK⁻ for the quadraticbaselinefits andthe linearbaseline fits.As itisusually the case in femtoscopicanalyses, the fitted R and λ parameters are correlated. The fitting (statistical) uncertainties are taken to be the extreme values of the 1

σ

fit contours in R vs. λ. Statis- ticaluncertaintiesareplottedforallresults.Itisseeninthefigure that the R and λ values for K⁰_SK⁻ have a slight tendency to be larger than those for K⁰_SK⁺. Such a difference could result from theK⁻–nucleonscatteringcrosssectionbeinglargerthanthatfor K⁺–nucleon(see Fig. 51.9 of Ref. [6]), possibly resulting inmore final-staterescatteringforthe K⁻.Since thedifferenceis notsig- nificantoncesystematicuncertaintiesaretakenintoaccount,K⁰_SK⁺ andK⁰_SK⁻ areaveraged overinthefinalresults.Thedifferencein theextracted parameters betweenthetwo baseline fittingmeth- odsisalsoseento besmall, andisaccountedforasa systematic error,asdescribedearlier.

TheresultsfortheR andλparametersextractedinthepresent analysis from K⁰_SK^± femtoscopy, averaged over the two baseline fitmethods andaveraged over K⁰_SK⁺ andK⁰_SK⁻,are presented in Table 2andinFigs. 4 and5.Fitresultsareshownforallfourpa- rametersetsgiveninTable 1.Figs. 4 and 5alsoshowcomparisons withidenticalkaonresultsforthesamecollision systemanden- ergyfromALICEfromRef.[5].Statisticalandtotaluncertaintiesare shownforallresults.

AsshowninFig. 4,bothAchasovparametersets,withthelarger a₀ massesanddecaycouplings,appeartogive Rvaluesthatagree bestwiththoseobtainedfromidentical-kaonfemtoscopy.TheAn-

tonelli parameter set appears to give slightlylower values.Com- paringthe measured R valuesbetweenK⁰_SK⁰_S andK^±K^± inFig. 4 theyareseentoagreewitheachotherwithintheuncertainties.In fact,theonlyreasonforthefemtoscopicK⁰_SK^±radiitobedifferent fromtheK⁰_SK⁰_SandK^±K^±oneswouldbeiftheK⁰_SandK^± sources weredisplacedwithrespecttoeachother.Thisisnotexpectedbe- causethecollisiondynamicsisgovernedbystronginteractionsfor whichtheisospinsymmetryapplies.

Theresultsforthecorrelationstrengthparametersλareshown in Fig. 5. The λ parameters fromK⁰_SK^± and K^±K^± are corrected forexperimental purity[5].The K⁰_SK⁰_S pairshavea highpurityof

>90%,sothecorresponding correctionwasneglected [5](seethe earlierdiscussiononpurity).Statistical andtotaluncertainties are shownforallresults.

The K⁰_SK^± λvalues,withtheexception oftheMartinparame- ters,appeartobeinagreementwiththeλvaluesfortheidentical kaons. All of the λ values are seen to be measured to be about 0.6, i.e. less than the ideal value of unity, which can be due to the contribution ofkaons fromK^∗ decay (∼^{50 MeV,where} isthedecaywidth)andfromotherlong-livedresonances(suchas the D-meson)distortingthespatialkaonsourcedistributionaway fromtheidealGaussianwhichisassumedinthefitfunction[26].

One wouldexpectthat theK⁰_SK^± λ valuesagreewiththosefrom theidenticalkaonsiftheFSIfortheK⁰_SK^±wentsolelythroughthe a0resonantchannelsincethisanalysisshouldseethesamesource distribution.

In order to obtain a more quantitative comparison of the present results for R and λ with the identical kaon results, the

χ

²/ndf iscalculatedforR andλforeachparameterset,

χ

_ω²

/

ndf

=

¹ ndf

3 i=¹

[ ω

i

(

K_S⁰K^±

) − ω

i

(

K K

) ]

²

σ

_i^{2 (12)}

Fig. 3.SampleresultsfortheRandλparametersextractedinthepresentanalysisfromK⁰_SK^±femtoscopyusingtheAchasov1parameters.TheleftcolumncomparesK⁰_SK⁺ andK⁰_SK⁻resultsfromthequadraticbaselinefitmethod,andtherightcolumncomparesresultsaveragedoverK⁰_SK⁺andK⁰_SK⁻forthequadraticbaselinefitsandthelinear baselinefits.Statisticaluncertaintiesareplottedforallresults.

Table 2

FitresultsforRandλextractedinthepresentanalysisfromK⁰_SK^± femtoscopyaveragedoverK⁰_SK⁺andK⁰_SK⁻. Statisticalandsystematicerrorsarealsoshown.

Parameters R(fm) orλ AllkT kT<0.675 GeV/c kT>0.675 GeV/c Achasov2 R 5.17±⁰.16±⁰.41 6.71±⁰.40±⁰.42 4.75±⁰.18±⁰.36

λ 0.587±⁰.034±⁰.051 0.651±⁰.073±⁰.076 0.600±⁰.040±⁰.034 Achasov1 R 4.92±0.15±0.39 6.30±0.40±0.43 4.49±0.18±0.30

λ 0.650±0.038±0.056 0.723±0.087±0.091 0.649±0.048±0.038 Antonelli R 4.66±⁰.17±⁰.46 5.74±⁰.36±⁰.26 4.07±⁰.18±⁰.29

λ 0.624±⁰.044±⁰.058 0.703±⁰.085±⁰.077 0.613±⁰.052±⁰.037 Martin R 3.29±⁰.12±⁰.35 4.46±⁰.25±⁰.20 2.90±⁰.11±⁰.41

λ 0.305±⁰.020±⁰.033 0.376±⁰.041±⁰.037 0.296±⁰.021±⁰.030

where

ω

iseitherRorλ,irunsoverthethreek_T values,thenum- berofdegreesoffreedomtakenisndf=^{3 and}

σ

iisthesumofthe statisticalandsystematicuncertainties onthe ithK⁰_SK^± extracted parameter(NotethattheallkT binindeedcontainsthekaonpairs that make up the kT<0.675 GeV/c and kT>0.675 GeV/c bins, butinadditionitcontainsanequalnumberofnewpaircombina- tionsbetweenthe kaonsinthe kT<0.675 GeV/c andkT>0.675 GeV/cbins.Soforthepurposesofthissimplecomparison,weap- proximate the all kT bin as being independent.) The linear sum ofthe statisticalandsystematicuncertaintiesisused for

σ

i tobe consistentwiththelinearsumofthestatisticalandsystematicun- certainties plotted on the points in Figs. 4 and 5. The quantity

ω

i(K K)isdeterminedby fittingaquadratictotheidenticalkaon resultsandevaluatingthefitattheaveragekT valuesoftheK⁰_SK^± measurements.Table 3summarizestheresultsforeachparameter set andthe extracted p-values. As seen, the Achasov2, Achasov1 andAntonelliparametersetsareconsistentwiththeidenticalkaon resultsforbothR andλ.TheMartinparametersetisseentohave vanishinglysmall p-valuesforboth R andλ and isthus inclear

Table 3

ComparisonsofRandλfromK⁰_SK^±withidenticalkaonresults.

Parameters χ_R²/ndf Rp-value χ_λ²/ndf λp-value

λ(K⁰_SK^±) λ(K K)

Achasov2 0.456 0.713 0.248 0.863 1.04±0.17

Achasov1 0.583 0.626 0.712 0.545 1.14±0.20

Antonelli 1.297 0.273 0.302 0.824 1.09±^0.20

Martin 14.0 0.000 22.2 0.000 0.55±^0.10

disagreementwiththeidenticalkaonresults,ascaneasilybeseen byexaminingFigs. 4 and 5.

Inordertoquantitativelyestimatethesizeofthenon-resonant channel present,the ratio

λ(K_S⁰K^±) λ(K K)

hasbeencalculatedforeach parameters set, where the average is over the three kT values andtheuncertaintyiscalculatedfromtheaverageofthestatisti- cal+systematicuncertaintiesontheK⁰_SK^±parameters.Thesevalues are showninthe lastcolumnofTable 3.Disregarding theMartin value,thesmallestvaluethisratiocantakewithin theuncertain-

Fig. 4.Sourceradiusparameter,R,extractedinthepresentanalysisfromK⁰_SK^±femtoscopyaveragedoverK⁰_SK⁺andK⁰_SK⁻andthetwobaselinefitmethods(redsymbols), alongwithcomparisonswithidenticalkaonresultsfromALICE[5](bluesymbols).Statistical(lines)andthelinearsumofstatisticalandsystematicuncertainties(boxes)are shown.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 5.Correlationstrengthparameter,λ,extractedinthepresentanalysisfromK⁰_SK^± femtoscopyaveragedoverK⁰_SK⁺ andK⁰_SK⁻ andthetwobaselinefitmethods(red symbols),alongwithcomparisonswithidenticalkaonresultsfromALICE[5](bluesymbols).Statistical(lines)andthelinearsumofstatisticalandsystematicuncertainties (boxes)areshown.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

ties is 0.87 (from the Achasov2 parameters) which would thus allowatmosta13%non-resonantcontribution.

Theresultsofthisstudypresentedaboveclearlyshowthatthe measuredK⁰_SK^± havedominantlyundergone aFSIthrough thea0 resonance.ThisisremarkableconsideringthatwemeasureinPb–

Pb collisions the average separation between the two kaons at freezeout to be ∼5 fm, and dueto the short-rangednature of thestrong interactionof ∼^{1 fmthis wouldseemto not encour-} agea FSIbutratherencouragefree-streamingofthekaonstothe detectorresultingina “flat” correlation function.A dominantFSI is what might be expected ifthe a0 would be a four-quark, i.e.

tetraquark,stateora“K–Kmolecule.”Thereappearstobenocal- culationsintheliteratureforthetetraquarkvs.diquarkproduction cross sections for the interaction KK→^a0, but qualitative argu- ments compatible with the a0 being a four–quark state can be madebasedonthepresentmeasurements.Themainargumentin favorofthisisthatthereactionchannelK⁰K⁻→^a−₀ ^(K0K+→^a+₀⁾ is strongly favored if the a⁻₀ (a⁺₀) is composed of dssu (dssu) quarkssuchthatadirecttransferofthequarksinthekaonstothe a⁻₀ (a⁺₀)hastakenplace,sincethisisan“OZIsuperallowed”reac- tion[12].The“OZIrule”canbestatedas“aninhibitionassociated withthecreationor annihilationofquark lines” [12].Thus, a di- quarka0 finalstateislessfavoredaccordingtotheOZIrulesince itwouldrequiretheannihilationofthestrangequarksinthekaon interaction. This would allow for the possibility of a significant non-resonant or free-streaming channel for the kaon interaction thatwouldresult ina λvalue belowthe identical-kaon value by dilutingthea₀signal.Asmentionedabove,thecollisiongeometry itself also suppresses the annihilation of the strange quarks due tothelargeseparationbetweenthekaonsatfreezeout.Notethat thisassumesthattheC(k^∗)distributionofanon-resonantchannel wouldbemostly “flat” or“monotonic”inshape andnotshowing astrongresonant-likesignalasseenforthea0 inFig. 1andFig. 2.

Thisassumption isclearly true inthe free-streaming case, which isassumedinEq.(9)in setting

α

=⁰.5 due tothe non-resonant kaoncombinations.Asimilarargument,namelythatthesuccessof the“chargedkaonloopmodel”indescribingtheradiativeφ-decay datafavorsthea0 asatetraquarkstate,isgiveninRef.[9].

5. Summary

In summary, femtoscopic correlations with K⁰_SK^± pairs have beenstudiedforthefirsttime.Thisnewfemtoscopicmethodwas appliedtodatafromcentral Pb–Pbcollisions at√

sNN=².76 TeV bythe LHCALICEexperiment.Correlations inthe K⁰_SK^± pairs are produced by final-state interactions which proceed through the a0(980) resonance.The a0 resonant FSI is seen to give an excel- lentrepresentationoftheshapeofthesignalregioninthepresent study.ThedifferencesbetweenK⁰K⁺andK⁰K⁻fortheextractedR andλvaluesarefoundtobeinsignificantwithintheuncertainties ofthepresentstudy.Thethreelargera0massanddecayparameter setsarefavoredbythecomparisonwiththeidenticalkaonresults.

Thepresentresultsare alsocompatiblewiththeinterpretationof thea0 resonance asatetraquark state. Thiswork should provide aconstraintonmodels that areusedto predictkaon–kaoninter- actions[27,28].Itwillbeinterestingtoapply K⁰_SK^± femtoscopyto other collision energies, e.g. the higher LHC energies now avail- able,andbombarding species, e.g. proton–protoncollisions, since the different source sizes encountered in these cases will probe theinteractionoftheK⁰_S withtheK^±indifferentsensitivityranges (i.e.seethe RdependenceinEq.(9)).

Acknowledgements

The ALICECollaboration wouldlike to thank all its engineers andtechniciansfortheirinvaluablecontributionstotheconstruc-

tion of the experiment and the CERN accelerator teams for the outstanding performance ofthe LHC complex.The ALICECollab- oration gratefully acknowledges the resources and support pro- videdbyallGridcenters andtheWorldwideLHCComputingGrid (WLCG) collaboration. The ALICE Collaboration acknowledges the followingfunding agenciesfortheir supportinbuildingandrun- ningtheALICEdetector:A.I.AlikhanyanNationalScienceLabora- tory(YerevanPhysicsInstitute)Foundation(ANSL),State Commit- teeofScienceandWorldFederationofScientists(WFS),Armenia;

Austrian AcademyofSciences andNationalstiftungfür Forschung, Technologie und Entwicklung, Austria; Ministry of Communica- tions and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Universidade Federal do Rio Grande do Sul (UFRGS), Financiadora de Estudos e Projetos (Finep) and Fun- dação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil; Ministry of Science & Technology of China (MSTC), Na- tional Natural Science Foundation of China (NSFC) and Ministry of Education of China (MOEC), China; Ministry of Science, Edu- cationandSports and Croatian ScienceFoundation, Croatia; Min- istryofEducation,YouthandSportsoftheCzech Republic,Czech Republic; The Danish Council for Independent Research Natu- ral Sciences, the Carlsberg Foundation and Danish National Re- search Foundation (DNRF), Denmark; HelsinkiInstitute of Physics (HIP),Finland;Commissariatàl’EnergieAtomique(CEA)andInsti- tut National de Physique Nucléaire et de Physique des Particules (IN2P3) andCentre Nationalde la Recherche Scientifique(CNRS), France; Bundesministerium für Bildung, Wissenschaft, Forschung undTechnologie (BMBF)andGSI Helmholtzzentrum für Schweri- onenforschung GmbH, Germany; GeneralSecretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary;DepartmentofAtomicEnergyGovernmentofIndia(DAE) andCouncilofScientificandIndustrialResearch(CSIR),NewDelhi, India; Indonesian Institute of Science, Indonesia; Centro Fermi – MuseoStorico dellaFisicae CentroStudi eRicerche EnricoFermi andIstitutoNazionalediFisicaNucleare(INFN),Italy;Institute for Innovative Science and Technology, Nagasaki Institute of Applied Science (IIST), Japan Society for the Promotion of Science (JSPS) KAKENHIandJapaneseMinistryofEducation,Culture, Sports,Sci- enceandTechnology (MEXT), Japan;Consejo Nacional de Ciencia y Tecnología(CONACYT), through Fondo de Cooperación Interna- cional enCienciay Tecnología(FONCICYT)andDirección General deAsuntosdelPersonalAcademico(DGAPA),Mexico;Nederlandse OrganisatievoorWetenschappelijkOnderzoek(NWO),Netherlands;

TheResearchCouncilofNorway,Norway;CommissiononScience andTechnology forSustainable Developmentin theSouth(COM- SATS),Pakistan;PontificiaUniversidadCatólicadelPerú,Peru;Min- istry ofScience and Higher Education andNational Science Cen- tre, Poland; Korea Institute of Science and Technology Informa- tion and National Research Foundation of Korea (NRF), Republic ofKorea;MinistryofEducationandScientificResearch,Instituteof Atomic Physicsand Romanian NationalAgency forScience, Tech- nology and Innovation, Romania; Joint Institute for Nuclear Re- search (JINR), Ministry of Education and Science of the Russian FederationandNationalResearchCentre KurchatovInstitute,Rus- sia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; NationalResearch Foundation of South Africa,South Africa;Centrode AplicacionesTecnológicasyDesar- rolloNuclear(CEADEN),Cubaenergía,Cuba,MinisteriodeCienciae InnovacionandCentrodeInvestigacionesEnergéticas,Medioambi- entalesyTecnológicas(CIEMAT),Spain;SwedishResearchCouncil (VR)andKnut&AliceWallenbergFoundation(KAW),Sweden;Eu- ropean Organization for Nuclear Research, Switzerland; National Science andTechnology Development Agency(NSDTA), Suranaree